The function $$ f $$ is defined by $$ f(x)=x^{3}+2 \sin (2 x+2) $$. If the function $$ g $$ is defined to be $$ g(x)=\int_{0}^{x} f(t) d t $$, what is the average rate of change of $$ g $$ on the interval $$ [-3,1] $$ ? You may use a calculator and round your answer to the nearest thousandth.

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To find the average rate of change of the function $$ g(x) = \int_{0}^{x} f(t) \, dt $$ on the interval $$[-3, 1]$$, follow these steps:

1. **Average Rate of Change Formula:**
   
   The average rate of change of $$ g $$ over the interval $$[a, b]$$ is given by:
   $$
   \text{Average Rate of Change} = \frac{g(b) - g(a)}{b - a}
   $$
   
   For $$ a = -3 $$ and $$ b = 1 $$:
   $$
   \text{Average Rate of Change} = \frac{g(1) - g(-3)}{1 - (-3)} = \frac{g(1) - g(-3)}{4}
   $$

2. **Evaluate $$ g(1) $$ and $$ g(-3) $$:**
   
   Since $$ g(x) = \int_{0}^{x} f(t) \, dt $$:
   $$
   g(1) = \int_{0}^{1} \left(t^3 + 2 \sin(2t + 2)\right) \, dt
   $$
   $$
   g(-3) = \int_{0}^{-3} \left(t^3 + 2 \sin(2t + 2)\right) \, dt = -\int_{-3}^{0} \left(t^3 + 2 \sin(2t + 2)\right) \, dt
   $$

3. **Combine the Integrals:**
   
   $$
   g(1) - g(-3) = \int_{0}^{1} \left(t^3 + 2 \sin(2t + 2)\right) \, dt + \int_{-3}^{0} \left(t^3 + 2 \sin(2t + 2)\right) \, dt = \int_{-3}^{1} \left(t^3 + 2 \sin(2t + 2)\right) \, dt
   $$

4. **Compute the Integral:**
   
   $$
   \int_{-3}^{1} t^3 \, dt = \left[\frac{t^4}{4}\right]_{-3}^{1} = \frac{1^4}{4} - \frac{(-3)^4}{4} = \frac{1}{4} - \frac{81}{4} = -20
   $$
   
   $$
   2 \int_{-3}^{1} \sin(2t + 2) \, dt = 2 \left[-\frac{\cos(2t + 2)}{2}\right]_{-3}^{1} = -\cos(4) + \cos(-4) = 0
   $$
   
   So, the total integral is:
   $$
   -20 + 0 = -20
   $$

5. **Calculate the Average Rate of Change:**
   
   $$
   \frac{-20}{4} = -5
   $$
   
   Rounding to the nearest thousandth:
   $$
   -5.000
   $$

**Final Answer:**  
The average rate of change of $$ g $$ on $$[-3, 1]$$ is $$-5.000$$.
The average rate of change of $$ g $$ on the interval $$[-3, 1]$$ is $$-5.000$$.

The function is defined by
. If the function is
defined to be , what is the average
rate of change of on the interval ? You may
use a calculator and round your answer to the nearest
thousandth.

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The function is defined by . If the function is defined to be , what is the average rate of change of on the interval ? You may use a calculator and round your answer to the nearest thousandth.